MULTIPLICATION OF VECTOR BY A SCALAR
A vector multiplied by a scalar λ results in another vector, λ. If λ is a positive number then λ is also in the direction of . If λ is a negative number, λ is in the opposite direction to the vector .
The scalar product (or dot product) of two vectors is defined as the product of the magnitudes of both the vectors and the cosine of the angle between them.
Thus if there are two vectors and having an angle θ between them, then their scalar product is defined as ⋅ = AB cos θ. Here, A and B are magnitudes of and .
The product quantity ⋅ is always a scalar. It is positive if the angle between the vectors is acute (i.e., < 90°) and negative if the angle between them is obtuse (i.e. 90°<θ< 180°).
The work done is basically a scalar product between the force vector and the displacement vector. Apart from work done, there are other physical quantities which are also defined through scalar products.
The vector product or cross product of two vectors is defined as another vector having a magnitude equal to the product of the magnitudes of two vectors and the sine of the angle between them. The direction of the product vector is perpendicular to the plane containing the two vectors, in accordance with the right hand screw rule or right hand thumb rule (Figure 2.22).
A number of quantities used in Physics are defined through vector products. Particularly physical quantities representing rotational effects like torque, angular momentum, are defined through vector products.
If two vectors and are equal, then their individual components are also equal.